Almost 1 million people have been displaced by escalating warfare in the two east Ukrainian regions of Donetsk and Luhansk, and hundreds of thousands of those have fled across the border to neighboring Russia, the United Nations said Tuesday.

Russian authorities say over 5,000 refugees per day are pouring into the southern Rostov region, which abuts the war zone, and that the numbers are putting strains on local facilities and infrastructure.

"It's getting close to a humanitarian crisis here," says Alexander Titov, spokesperson for Rostov Governor Vasily Golubev. "Depending on the fighting on any given day, between five and ten thousand people come across the border. We have 95 temporary refugee centers – basically tent camps – operating now, up from 15 in June. We try to send people on to other regions as fast as we can, but the numbers are still swelling.

"We're very alarmed, because cold weather is just around the corner, and those tents aren't heated," Mr. Titov says.

As Ukrainian forces tighten a steel noose around the rebel-held cities of Donetsk and Luhansk, the numbers of civilians desperately trying to escape the war zone is spiking. Credible reports have documented the widespread use of heavy artillery and unguided rockets against civilian targets by both sides in the conflict.

Before the war, about 6.5 million people lived in Donetsk and Luhansk, or around 14 percent of Ukraine's population. The few reports coming out of those places now show widespread ruination and empty streets. A report by the Kremlin-funded RT network from Luhansk on Tuesday said that about a quarter million inhabitants were still trapped in the city, without power or water, with dwindling food supplies and facing constant shellfire from outside the city.

Citing Russian authorities, the UNHCR said that 730,000 Ukrainians have entered Russia this year, and that 168,000 of those had applied for temporary or permanent asylum in Russia.

"We don't call all of those [730,000] people refugees," Vincent Cochetel, head of the UNHCR's European Bureau, told journalists in Geneva. "But they are not tourists. We have seen them at the border... Sometimes they just walk across the border, some just with plastic bags, and many of them are really destitute."

A further 117,000 people have fled the war zone to other parts of Ukraine, he added.

The head of Russia's Federal Migration Service, Konstantin Romodanovsky, said Tuesday that there are 2 million Ukrainians currently living in Russia, about 600,000 of whom are from Donetsk and Luhansk. He added that Russian authorities expect about 90 percent of them to return to their homes once the fighting ends.

Many refugees find temporary accommodation with relatives or other Russian families, and there is now a major effort in Rostov region to ensure that thousands of Ukrainian children will be enrolled in Russian kindergartens and schools by the beginning of September, Titov says.

"We've managed to get volunteers into the refugee camps to cook, and help out. In most we have water and hot showers. About 100 women have given birth [in the camps] already," he says. "But they keep coming, and we're making big efforts to send them on to other regions too."